Let $l^p$ be the space of $p$-summable sequences. von Neumann constructed a subset of $l^p$ space
$$S=\{X_{mn}: m,n≥1\}$$
where $X_{mn}\in l^p$ are defined by $X_{mn}(m)=1, X_{mn}(n)=m$ and $X_{mn}(k)=0$ otherwise.

I am asked to show that this $S$ is closed in the strong topology. I tried to show the complement is open by trying to construct a contradiction, but no success. Could anyone help me ? Thanks in advance.

I dont know how to prove 1 . But following the comment , any cauchy sequnce in such a space must eventually be constant and thus converges in S. The space should be complete and thus closed . Just wondering how to show 1 by topological argument ?
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needhelpOct 30 '12 at 15:12